15  Notation

15.1 Random variables, vectors, and matrices

• Random variables are capital letters such as $X$ and $Y$.
• A particular value of $X$ is $x$.
• Vectors are bold lowercase symbols (e.g., $\boldsymbol{a}$, $\boldsymbol{\theta}$) or italic lowercase with arrows (e.g., $\vec{a}$, $\vec{\theta}$).
• Matrices are bold uppercase symbols (e.g., $\boldsymbol{A}$, $\boldsymbol{\Theta}$).

Expression	Meaning
$\boldsymbol{A'}$	The transpose of matrix $\boldsymbol{A}$
$\boldsymbol{A}^{-1}$	The inverse of matrix $\boldsymbol{A}$
$\mathtt{diag}(\boldsymbol{A})$	The vector formed from the diagonal of matrix $\boldsymbol{A}$
$\mathtt{diag}(\boldsymbol{a})$	The diagonal matrix formed from vector $\boldsymbol{a}$
$\boldsymbol{1}_k$	A column vector of $k$ ones. If $k$ is absent, the length of $\boldsymbol{1}$ can be inferred by context.
$\boldsymbol{I}_k$	A $k \times k$ identity matrix. If $k$ is absent, the dimensions of $\boldsymbol{I}$ can be inferred by context.

15.2 Sets and intervals

A set is a collection of distinct objects.

Expression	Meaning
$\{a,b\}$	A set consisting of two numbers, $a$ and $b$
$\lbrack a,b\rbrack$	The set of real numbers in the interval between $a$ and $b$, inclusive
$(a,b)$	The set of real numbers in the interval between $a$ and $b$, exclusive
$(a,b\rbrack$	The set of real numbers in the interval between $a$ and $b$, excluding $a$ but including $b$
$\mathbb{R}$	The set of all real numbers $[-\infty,\infty]$
$\mathbb{Z}$	The set of all integers $\{\ldots,-2,-1,0,1,2,\ldots\}$
$\mathbb{N}_0$	The set of all non-negative integers: $\{ 0,1,2,...\}$
$\mathbb{N}_1$	The set of all positive integers: $\{ 1,2,3,...\}$
$\in$	Is a member of (i.e., $X \in (a,b)$ means that $X$ is a member of the set of numbers between $a$ and $b$)

15.3 Summation

There are several ways to show that a variable is to be summed. The summation sign looks like a really big Σ, the capital Greek letter sigma. If $\boldsymbol{x}$ is a vector with $k$ elements, the sum of all $k$ in $\boldsymbol{x}$ is:

$$\sum_{i = 1}^k x_i$$

With matrix algebra,

$$\sum_{i = 1}^k x_i=\boldsymbol{1}'\boldsymbol{x}$$

15.4 Statistics

Expression	Meaning
$\mu_X, m_X$	The population and sample mean of $X$
$\sigma_X, s_X$	The population and sample standard deviation of $X$
$\sigma_X^2, s_X^2$	The population and sample variance of $X$
$\gamma_1, g_1$	The population and sample skewness
$\gamma_2, g_2$	The population and sample kurtosis

15.5 Other

Expression	Meaning
$\binom{n}{k}$	The binomial coefficient. It is just a shortcut notation for $\binom{n}{k}=\frac{n!}{k!\left(n-k\right)!}$. Read aloud, $\binom{n}{k}$ is ``$n$ choose $k$’’ or the number of combinations that $n$ things have when taken $k$ at a time.
$f_X(x;\boldsymbol{\theta})$	The probability density function or probability mass function of $X$ with parameters $\boldsymbol{\theta}$
$F_X(x;\boldsymbol{\theta})$	The cumulative distribution function of $X$ with parameters $\boldsymbol{\theta}$
$\mathcal{E}\left(X\right)$	The expected value of $X$